The FSA warns that more than 20,000 Britons a year die prematurely because of the high level of salt in their diet.
But food producers claimed that the new requirements would affect taste while bakers warned that it was "technically impossible" for them to meet the 2012 targets.
They include recommendations to reduce the amount of salt in hamburgers by another quarter on top of the cuts the organisation has already demanded be made by 2010. By 2010 burgers should contain no more than 1g of salt per 100g and by 2012 no more than 0.75g per 100g.
The FSA wants every adult in England to eat no more than 6g of salt a day.
But a recent study of nearly 700 British adults showed that average salt intake was 8.6g a day. Average consumption of salt has fallen from 9.5g a day in 2001 and 9g last year.
The agency warns that much of our salt intake is "hidden" in everyday foods.
Producers of bacon, sausages, cheese, butter, breakfast cereals, baked beans and ready meals will all have to make further cuts in the amount of salt they use to meet the new targets, described by the FSA as "challenging".
But industry groups warn that the speed at which they would be forced to make the changes mean that consumers could notice a detrimental effect on taste.
Stephen Robertson, director general of the British Retail Consortium (BRC), warned that if changes happened too quickly people would simply compensate by adding salt to their food at home.
"It's crucial we take customers with us, as tastes don't change overnight," he said.
The Federation of Bakers warned that its members were already in the process of cutting salt levels as low as they could go, to meet the 2010 targets.
Gordon Polson, the organisation's director, said he was "concerned".
"It is technically impossible for the industry to go beyond the 2010 target," he added.
But Mike Rich, executive director of the Blood Pressure Association, claimed that some of the proposed original targets for salt levels in bread had been relaxed, a situation he described as "disappointing".
If the target of 6g a day was met, some 20,200 premature deaths a year could be prevented from conditions like stroke, heart attacks and heart failure, the FSA estimate.
The agency praised "considerable reductions" in salt levels that have been made by manufacturers but it said there was clearly room for "some parts of industry to do more".
Last year the organisation toughened its some of the targets it had already set for 2010 for the 80 different types of food it monitors.